Codeforces 939E - Maximize! 三分

最新推荐文章于 2022-07-13 00:03:04 发布

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最新推荐文章于 2022-07-13 00:03:04 发布

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本文解析了CodeForces上的一道题目E.Maximize!，介绍了如何通过三分法找到最大值的问题解决思路，并给出了完整的AC代码实现。

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http://codeforces.com/problemset/problem/939/E

题目：

E. Maximize!

time limit per test

3 seconds

memory limit per test

256 megabytes

input

standard input

output

standard output

You are given a multiset S consisting of positive integers (initially empty). There are two kind of queries:

Add a positive integer to S, the newly added integer is not less than any number in it.
Find a subset s of the set S such that the value $\text{[math]}$ is maximum possible. Here max(s) means maximum value of elements in s, $\text{[math]}$ — the average value of numbers in s. Output this maximum possible value of $\text{[math]}$ .

Input

The first line contains a single integer Q (1 ≤ Q ≤ 5·105) — the number of queries.

Each of the next Q lines contains a description of query. For queries of type 1 two integers 1 and x are given, where x (1 ≤ x ≤ 109) is a number that you should add to S. It's guaranteed that x is not less than any number in S. For queries of type 2, a single integer 2 is given.

It's guaranteed that the first query has type 1, i. e. S is not empty when a query of type 2 comes.

Output

Output the answer for each query of the second type in the order these queries are given in input. Each number should be printed in separate line.

Your answer is considered correct, if each of your answers has absolute or relative error not greater than 10 - 6.

Formally, let your answer be a, and the jury's answer be b. Your answer is considered correct if $\text{[math]}$ .

Examples

Input

Output

0.0000000000
0.5000000000
3.0000000000

Input

Output

2.0000000000

题意：

第一行是一个Q，表示Q次操作。

两种操作操作一：在集合S中添加元素x，保证x不小于S中任何一个元素（保证S递增）；操作二：查询，s为S子集，求函数max(s)-mean(s)的最大值，max(s)表示s中元素的最大值，mean(s)表示s中所有元素的平均数。对于每次操作二，输出对应函数最大值，误差小于10e-6。

n为S元素个数，由于S递增，要使max(s)-mean(s)最大，只需要取S中的最后一个元素和S中前m(m<=n-1)个元素即可，而且由1到m函数max(s)-mean(s)的值先增加后减小，可以使用三分来求最值

三分原理：

lmid和rmid是l到r的两个三等分点，如果lmid大于rmid，那么最大值max一定在l与rmid之间，如果lmid小于rmid那么最大值max一定在lmid与r之间。

AC代码：

#include<stdio.h>
#include<string.h>
#include<math.h>
#include<algorithm>
using namespace std;
long long a[500005],a_sum[500005];
int m;
double f(int x)
{
    return (double)a[m]-((a_sum[x]+a[m])/(double)(x+1));
}
int main()
{
    int n,i,j,x,l,r,lmid,rmid;
    double Max;
    a_sum[0]=0;
    m=0;
    scanf("%d",&n);
    for(i=1; i<=n; i++)
    {
        scanf("%d",&x);
        if(x==1)
        {
            m++;
            scanf("%d",&a[m]);
            a_sum[m]=a_sum[m-1]+a[m];
        }
        else
        {
            l=1;
            r=m-1;
            while(r-l>2)
            {
                lmid=l+(r-l)/3;
                rmid=r-(r-l)/3;
                if(f(lmid)>f(rmid))
                {
                    r=rmid;
                }
                else if(f(lmid)<f(rmid))
                {
                    l=lmid;
                }
                else
                {
                    l=lmid;
                    r=rmid;
                }
            }
            Max=0;
            for(j=l; j<=r; j++)
                Max=max(Max,f(j));
            printf("%.8lf\n",Max);
        }
    }
    return 0;
}